In this article, we investigate the following delay differential equation $$ (\triangle_c \nu(z))^{2}-\tilde{A}(z)\nu(z)\nu(z+c)+\tilde{B}(z)\frac{\nu’(z)}{\nu(z)}=R(z, \nu(z)), $$ where $R(z, \nu(z))$ be an irreducible rational function in $\nu(z)$ with rational coefficients, $\tilde{A}(z)$ and $\tilde{B}(z)$ be a rational function. We give necessary conditions on the degree of $R(z, \nu)$ for the above equation to admit a transcendental entire solution $\nu(z)$ with $\sigma_{2}(\nu)<1$. In addition, for certain classes of equations, we prove the existence of rational solutions and give their forms.